Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds

Interpretable Stein Goodness-of-fit Tests on Riemannian Manifolds

In many applications, we encounter data on Riemannian manifolds such as torus and rotation groups. Standard statistical procedures for multivariate data are not applicable to such data. In this study, we develop goodness-of-fit testing and interpretable model criticism methods for general distributions on Riemannian manifolds, including those with an intractable normalization constant. The proposed methods are based on extensions of kernel Stein discrepancy, which are derived from Stein operators on Riemannian manifolds. We discuss the connections between the proposed tests with existing ones …

Goodness-of-fit testing and quadratic functional estimation from indirect observations

We consider the convolution model where i.i.d. random variables $X_i$ having unknown density $f$ are observed with additive i.i.d. noise, independent of the $X$'s. We assume that the density $f$ belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional $\int f^2$ based o…

来自乘法测量误差观测的拟合检验

Goodness-of-fit testing from observations with multiplicative measurement error

本文提出了一种基于 Mellin 变换二次泛函的非参数拟合检验方法，用于处理乘法测量误差下的密度检验问题，并通过模拟研究验证了其有效性。

Goodness-of-fit test for noisy directional data

We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \Cψ_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $ν$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $ψ_N^{ad}(s)=(N/\sqrt{\log\log…

KSD 聚合拟合优度检验

KSD Aggregated Goodness-of-fit Test

本文提出 KSDAgg 聚合检验，通过聚合多个核的 KSD 检验并避免数据分割，实现了最优功效，并在理论和实验中优于现有方法。

Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs

In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator $\widehat{f}$, and a goodness-of-fit test also based on $\widehat{f}$. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X}\subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Ad…